Nonlinear Physics

A course I much enjoyed this semester was on the equations of mathematical physics (PHYS384 for those at SFU.) We learnt a whole bunch of methods to solve and analyze equations that fit into the Sturm-Liouville framework of linear partial differential equations. The natural next step is to extend this body of knowledge to non-linear equations of mathematical physics. That is exactly what I’m doing next term.

We have a whole bunch of methods to analyze and predict the performance of linear systems, but systems in nature aren’t always that idealistic. The classic example is that of a frictionless pendulum with the equation of motion:

$\ddot{\theta} + \omega_0^2 \sin(\theta) = 0$

where $\omega_0 = \sqrt{(g/l)}$ , g is the acceleration due to gravity and l is the length of the pendulum. The sine function being non-linear is often linearized to just $\theta$ for small angles. This is the standard harmonic oscillator equation.

The real challenge is in analyzing the original non-linear equation. The difficulty in solving this equation is readily seen in the following Maxima session, where it isn’t smart enough to recognize the Jacobian elliptical integral.

Maxima session

The only course I’m taking next term is titled “Instabilities, Chaos and Turbulence in non-linear and complex systems” numbered PHYS484. We’ll explore a lot of these concepts. I’m really looking forward to it. Also check out this really cool experiment with a liquid:

This entry was posted on Sunday, December 17th, 2006 at 8:10 pm and is filed under Physics. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.

3 Responses to “Nonlinear Physics”

Looking ahead » Ganesh Swami says:

December 31, 2006 at 7:51 pm

[...] As I’m only taking only one course next term, you might be wondering what I’m upto. I’ll be working at the Medical Imaging Analysis Lab (MIAL) for the next eight months. To be specific, I’ll be working in the field of Computational Anatomy, which is “the use of mathematical analysis to learn how tissues grow, assume new shapes and morph into mature structures.” [...]
Phase space » Ganesh Swami says:

January 14, 2007 at 8:14 pm

[...] Each point on the phase space acts as an initial condition for the system. As I had written earlier, solving this equation is not trivial. If instead we look at the phase plane, we can determine three distinct regions of operations: [...]
Faraday crispations » Ganesh Swami says:

February 24, 2007 at 2:20 am

[...] Faraday waves were first described in an appendix to a paper published in the Philosophical Transactions of the Royal Society of London in 1831. These are standing non-linear waves that are generated when an open container with fluid is subject to vertical oscillations. When the oscillations reach a certain threshold, we begin to see an instability on the surface of the fluid. Our professor did a demo for us with two fluids: canola and water. I had posted a video to Faraday waves with corn starch some time back. [...]